JOHNSON SU DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.JohnsonSU({"xi": *, "lambda": *, "gamma": *, "delta": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

## CDF, PDF, PPF receive float or numpy.ndarray.
distribution.cdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.pdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.ppf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.sample(int) # -> numpy.ndarray

## STATS
distribution.mean # -> float
distribution.variance # -> float
distribution.standard_deviation # -> float
distribution.skewness # -> float
distribution.kurtosis # -> float
distribution.median # -> float
distribution.mode # -> float

Equations

Distribution Definition

$$X\sim \mathrm{JohnsonSU}(\xi ,\lambda ,\gamma ,\delta )$$

Distribution Domain

$$x\in (-\mathrm{\infty },\mathrm{\infty })$$

Parameters Domain and Constraints

$$\xi \in \mathbb{R},\lambda \in {\mathbb{R}}^{+},\gamma \in \mathbb{R},\delta \in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\mathrm{\Phi }(\gamma +\delta {\sinh }^{-1}(z(x)))$$

Probability Density Function

$$f_X\left(x\right)=\frac{\delta }{\lambda \sqrt{2\pi }\sqrt{z(x{)}^2+1}}\exp [-\frac{1}{2}{(\gamma +\delta {\sinh }^{-1}(z(x)))}^2]$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\lambda \sinh \left(\frac{{\mathrm{\Phi }}^{-1}(u)-\gamma }{\delta }\right)+\xi$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx$$

Parametric Mean

$$\mathrm{Mean}(X)={\mu }_1^{\prime }=\xi -\lambda \exp \frac{{\delta }^{-2}}{2}\sinh \left(\frac{\gamma }{\delta }\right)$$

Parametric Variance

$$\mathrm{Variance}(X)={\mu }_2^{\prime }-{\mu }_1^{\prime 2}=\frac{{\lambda }^2}{2}(\exp ({\delta }^{-2})-1)(\exp ({\delta }^{-2})\cosh \left(\frac{2\gamma }{\delta }\right)+1)$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=-\frac{{\lambda }^3\sqrt{e^{{\delta }^{-2}}}(e^{{\delta }^{-2}}-1{)}^2(e^{{\delta }^{-2}})(e^{{\delta }^{-2}}+2)\sinh (\frac{3\gamma }{\delta })+3\sinh (\frac{2\gamma }{\delta }))}{4\mathrm{Variance}(X{)}^{1.5}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=\frac{{\lambda }^4(e^{{\delta }^{-2}}-1{)}^2(K_1+K_2+K_3)}{8\mathrm{Variance}(X{)}^2}$$

Parametric Median

$$\mathrm{Median}(X)=\xi +\lambda \sinh (-\frac{\gamma }{\delta })$$

Parametric Mode

$$\mathrm{Mode}(X)=\arg \underset{x}{max}{f}_X\left(x\right)$$

Additional Information and Definitions

• $\xi :\text{Location parameter}$
• $\lambda :\text{Scale parameter}$
• $z\left(x\right)=(x-\xi )/\lambda$
• $u:\text{Uniform[0,1] random varible}$
• $\mathrm{\Phi }\left(x\right):\text{CDF normal standard distribution}$
• ${\mathrm{\Phi }}^{-1}\left(x\right):\text{PPF normal standard distribution}$
• $K_1={\left(e^{{\delta }^{-2}}\right)}^2({\left(e^{{\delta }^{-2}}\right)}^4+2{\left(e^{{\delta }^{-2}}\right)}^3+3{\left(e^{{\delta }^{-2}}\right)}^2-3)\cosh \left(\frac{4\gamma }{\delta }\right)$
• $K_2=4{\left(e^{{\delta }^{-2}}\right)}^2(\left(e^{{\delta }^{-2}}\right)+2)\cosh \left(\frac{3\gamma }{\delta }\right)$
• $K_3=3(2\left(e^{{\delta }^{-2}}\right)+1)$

Spreadsheet Documents

• Phitter playground
• Download Excel Spreadsheet
• Excel file from GitHub repository
• Google spreadsheet document